Random Geometric Complexes

Abstract

A survey of several results from the recently flourishing field of random geometric complexes, a field combining combinatorics, probability theory and topology. Random geometric complexes are a higher dimensional analogue of random geometric graphs; they are combinatorially described topological spaces, built using random geometric information, i.e. points and distances between these points. The studied results relate to the Betti numbers (number of holes) of random geometric complexes in different regimes (number of points expected per volume). The proofs of these results are given in a lot of detail, so that any student or researcher aspiring to enter the field can use the thesis as a starting point.